Page:LorentzGravitation1916.djvu/58

Each value of occurring twice, i.e. combined with the two values different from which can take, we have in addition to (118)

so that (117) becomes

As now outside the sphere

we have for every closed surface that does not surround the sphere , but for every surface that does

(119)

As to we remark that substituting (65) in (41) and taking into consideration (64) we find,

(120)

From this we conclude that is zero if there is no matter inside the surface . In order to determine in the opposite case, we remember that is independent of the choice of coordinates. To calculate this quantity we may therefore use the value of indicated in § 56, which is sufficient to calculate as far as the terms of the first order. We have therefore

and if, using further on rectangular coordinates, we take for the normal value ,

From this we find by substitution in (114) for the case of the closed surface a surrounding the sphere

This equation together with (119) shows that in (113) when integrated over the whole space the terms of the first order really cancel each other. In order to calculate those of the second order